Anything which can be performed on basic mathematical operations like addition, subtraction, multiplication and division is called a quantity.

**Functions and their graphs**

**Some basic concepts**

Anything which can be performed on basic mathematical operations like addition, subtraction, multiplication and division is called a quantity.

A quantity which retains the same value throughout a mathematical investigation is called a constant.

Basically constant quantities are of two types

** (i). Absolute constants **are those which do not change their values in any** **mathematical investigation. In other words, they are fixed for ever.

**Examples: **3, √3 , p, ...

**(ii) Arbitrary constants **are those which retain the same value throughout a** **problem, but we may assign different values to get different solutions. The arbitrary constants are usually denoted by the letters *a, b, c,* ...

**Example: **In an equation** ***y=mx+4*,** ***m*** **is called arbitrary constant.

A variable is a quantity which can assume different values in a particular problem. Variables are generally denoted by the letters *x, y, z, ..*.

**Example: **In an equation of the straight line

*x *and* y *are variables because they assumes the co-ordinates of a moving point in a* *straight line and thus changes its value from point to point. *a* and *b* are intercept values on the axes which are arbitrary.

There are two kinds of variables

i. A variable is said to be an **independent variable** when it can have any arbitrary value.

ii. A variable is said to be a **dependent variable** when its value depend on the value assumed by some other variable.

**Example: **In the equation y =** **5*x*2** **-** **2*x*** **+** **3,

*“x” *is the independent variable,

*“y” *is the dependent variable and

*“*3*” *is the constant.

The real numbers can be represented geometrically as points on a number line called real line. The symbol *R* denotes either the real number system or the real line. A subset of the real line is an interval. It contains atleast two numbers and all the real numbers lying between them.

L et X and Y be two non-empty sets of real numbers. If there exists a rule f which associates to every element x Î X, a unique element y Î Y, then such a rule f is called a function (or mapping) from the set X to the set Y. We write f : X -- >Y.

The set X is called the domain of f, Y is called the co-domain of f and the range of f is defined as f(X) = {f(x) / x Î X}. Clearly f(X) ÍY.

A function of x is generally denoted by the symbol f(x), and read as* “function of x” or “f of x”.*

Functions can be classified into two groups.

**i. Algebraic functions**

Algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division and raising to a fractional power.

A function *f(x)* is said to be an even function of *x,* if *f(* - *x ) = f( x ).*

A function *f(x)* is said to be an odd function of *x,* if *f(* - *x ) =* - *f( x ).*

A function in which the dependent variable is expressed explicitly in terms of some independent variables is known as **explicit function**.

**Examples: ***y = x2*** ***+ 3*** **and** ***y = ex*** ***+ e*-*x*** **are explicit functions of** ***x.*

If two variables *x* and *y* are connected by the relation or function *f*(*x*, *y*) = 0 and none of the variable is directly expressed in terms of the other, then the function is called an **implicit function**.

**Example: ***x3*** ***+ y3*-*xy = 0*** **is an implicit function.

If *k* is a fixed real number then the function given by* f*(*x*) =* k *for all* x *Î* R *is called a constant function.

**Examples: ***y = 3, f*(*x*) = –5 are constant functions.

A function that associates each real number to itself is called the identity function and is denoted by I.

i.e. A function defined on *R* by *f*(*x*) = *x* for all *x *∈* R *is an identity function.

**Example: **The set of ordered pairs {(1, 1), (2, 2),** **(3, 3)} defined by *f : A* -> *A* where *A*={1, 2, 3} is an identity function.

The function whose value at any real number *x* is the greatest integer less than or equal to *x* is called the greatest integer function. It is denoted by *x* .

i.e. *f* : *R* -> *R* defined by *f*(*x*) = ë*x*û is called the greatest integer function.

The function whose value at any real number *x* is the smallest integer greater than or equal to *x* is called the least integer function. It is denoted by *x* .

i.e. *f* : *R* à *R* defined by *f*(*x*) = é *x* ù is called the least integer function

**Remark : Function’s domain is R and range is Z [set of integers]**

For the real numbers *a*0, *a*1, *a*2, ..., *an* ; *a*0 ¹ 0 and *n* is a non-negative integer, a function *f*(*x*) given by *f*] *x*g = *a*0 *xn* + *a*1 *xn* - 1 + *a*2 *xn* - 2 + ... + *an* is called as a polynomial function of degree *n*.

**Example: ***f*** **(*x*** **)** **=** **2*x*** **3** **+** **3*x*2** **+** **2*x*** **-** **7** **is a polynomial function of degree 3.

For the real numbers *a* and *b* with a ¹ 0, a function *f*(*x*) = *ax* + *b* is called a linear function.

**Example: ***y =*** **2*x*** **+ 3 is a linear function.

For the real numbers *a, b* and *c* with *a* ¹ 0, a function *f*(*x*) = *ax*2 + *bx* + *c* is called a quadratic function.

**Example: ***f*** **(*x*** **)** **=** **3*x*2** **+** **2*x*** **-** **7** **is a quadratic function.

A function *f*(*x*) = *ax*, *a* ¹ 1 and *a* > 0, for all *x* ∈ *R* is called an exponential function

**Remark **: Domain is R and range is (0, ∞) and (0, 1) is a point on the graph

**Examples: ***e*2*x*** **,** ***ex*2** **+** **1** **and** **2*x*** **are exponential functions.

For *x* > 0, *a* > 0 and *a* ¹1, a function *f*(*x*) defined by *f*(*x*) = log*ax* is called the logarithmic function.

**Remark:** Domain is (0, ∞), range is R and (1, 0) is a point on the graph

**Example: ***f*(*x*) = log*e*(*x*+2),** ***f*(*x*) = log*e*** **(sin*x*) are logarithmic functions.

T he graph of a function is the set of points (x , f(x)) where x belongs to the domain of the function and f(x) is the value of the function at x.

To draw the graph of a function, we find a sufficient number of ordered pairs (x , f(x)) belonging to the function and join them by a smooth curve.

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